Quadratically Dense Matroids
| dc.contributor.author | Walsh, Zachary | |
| dc.date.accessioned | 2020-07-08T17:35:10Z | |
| dc.date.available | 2020-07-08T17:35:10Z | |
| dc.date.issued | 2020-07-08 | |
| dc.date.submitted | 2020-07-07 | |
| dc.description.abstract | This thesis is concerned with finding the maximum density of rank-$n$ matroids in a minor-closed class. The extremal function of a non-empty minor-closed class $\mathcal M$ of matroids which excludes a rank-2 uniform matroid is defined by $$h_{\mathcal M}(n)=\max(|M|\colon M\in \mathcal M \text{ is simple, and } r(M)\le n).$$ The Growth Rate Theorem of Geelen, Kabell, Kung, and Whittle shows that this function is either linear, quadratic, or exponential in $n$. In this thesis we prove a general result about classes with quadratic extremal function, and then use it to determine the extremal function for several interesting classes of representable matroids, for sufficiently large integers $n$. In particular, for each integer $t\ge 4$ we find the extremal function for all but finitely many $n$ for the class of $\mathbb C$-representable matroids with no $U_{2,t}$-minor, and we find the extremal function for the class of matroids representable over finite fields $\mathbb F_1$ and $\mathbb F_2$ where $|\mathbb F_1|-1$ divides $|\mathbb F_2|-1$ and $|\mathbb F_1|$ and $|\mathbb F_2|$ are relatively prime. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/16051 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | matroids | en |
| dc.subject | combinatorics | en |
| dc.subject | density | en |
| dc.subject | quadratic | en |
| dc.subject | extremal | en |
| dc.subject | fields | en |
| dc.title | Quadratically Dense Matroids | en |
| dc.type | Doctoral Thesis | en |
| uws-etd.degree | Doctor of Philosophy | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws-etd.degree.discipline | Combinatorics and Optimization | en |
| uws-etd.degree.grantor | University of Waterloo | en |
| uws.comment.hidden | Can my abstract be typeset with LaTex? | en |
| uws.contributor.advisor | Nelson, Peter | |
| uws.contributor.affiliation1 | Faculty of Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.published.city | Waterloo | en |
| uws.published.country | Canada | en |
| uws.published.province | Ontario | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |